physics-ai999x — offline, verified physics solver with step-by-step answers

Install · Capability guide · ONNX models · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 2.0.0

physics-ai999x is an offline, deterministic Python solver that answers physics questions with verified results from 148 certified capabilities, gives 859 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

physics-ai999x is a compact, deterministic physics engine for Python. It reads a question in plain English (or a typed call), picks a certified physics capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified physics capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Measured: 0 wrong answers on 274 benchmark questions and on 1,567 verified natural-language questions.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• ONNX included. 365 deterministic formula graphs (onnx/formulas/, 94.7 KB) + AI999X capability encoder (onnx/search/encoder.int8.onnx, 4.9 MB int8) + decision model (onnx/decision.int8.onnx, 37.4 MB int8): run the exact formulas without Python — onnxruntime in C++, C#, Java, JavaScript or the browser; 13,256 of 13,256 checks agree with the Python engine.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: kinematics, forces and Newton's laws, work and energy, momentum and collisions, circular motion and gravitation, rotation, oscillations and waves, fluids, heat and thermodynamics, electricity and circuits, magnetism and induction, optics and modern physics.

Model overview

Property	Value
Package	physics-ai999x 2.0.0 (import physics_ai999x)
Type	Deterministic neuro-symbolic physics engine (rule-based reading + certified symbolic capabilities)
Capabilities	23 certified derivations, 16 certified model_familys, 2 certified procedures, 107 certified relations
Knowledge pack	version 0.6.0
Golden answers	859 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
ONNX	365 deterministic formula graphs (onnx/formulas/, 94.7 KB) + AI999X capability encoder (onnx/search/encoder.int8.onnx, 4.9 MB int8) + decision model (onnx/decision.int8.onnx, 37.4 MB int8)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install physics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/physics-ai999x/resolve/main/dist/physics_ai999x-2.0.0-py3-none-any.whl

import physics_ai999x as physics

physics.ask("A 1000 kg car moving at 20 m/s hits a stationary 500 kg car and they stick together. What is their final velocity?").show()
# -> SOLVED — 13.3333 m/s  (collision_1d)
physics.ask("How much heat boils away 0.5 kg of water? Its latent heat of vaporization is 2260000 J/kg.").show()
# -> SOLVED — 1.13e+06 J  (latent_heat_vaporization)
physics.ask("A hockey puck slides across ice at 10 m/s and slows down at a constant rate of 0.5 m/s^2. What is its speed after 8 seconds?").show()
# -> SOLVED — 6 m/s  (velocity_time)
physics.solve("kinetic_energy", mass="3 kg", speed="4 m/s").show()
# -> SOLVED — 24 J  (kinetic_energy)
physics.ask("What is the kinetic energy of a car moving at 20 m/s?").show()
# -> NEEDS_INPUT — Translational kinetic energy applies, but it needs: mass (kg). Add it to the question with its unit, or use so
physics.ask("Who created you?").message
# -> physics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: physics-ai999x "<question>", physics-ai999x --capabilities. The outputs above were produced by running this exact code on physics-ai999x 2.0.0.

Find the right capability, and draw it

physics.search() finds which certified capability a question is about — across every AI999X model, so a question for another model points you to it. It runs ai999x-encoder, AI999X's own capability encoder (4.7M parameters, a 4.9 MB ONNX model, offline; the root model.safetensors of this repository): pip install "physics-ai999x[search]". ask(question, suggest=True) adds the same pointers to a question it does not answer; an answer itself never changes.

import physics_ai999x as physics

physics.search("How long does a pendulum take to swing back and forth?", k=2)
# -> Period of a simple pendulum (physics-ai999x, 0.78); Precession period of a Foucault pendulum (mechanics-ai999x, 0.68)
physics.search("Solve x^2 - 5x + 6 = 0", k=2)
# -> Roots of a quadratic equation (algebra-ai999x, 0.67); Solution of two linear equations in two unknowns (algebra-ai999x, 0.57)

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Benchmarks

Evaluation set	Questions	Correct	Refused (with reason)	Wrong
Development probes	112	68	44	0
Held-out: fresh questions (2026-10-15)	42	32	10	0
Held-out: probes	34	30	4	0
Held-out: promotion checks (2026-10-13)	27	14	13	0
Held-out: multi-body systems (2026-10-15)	43	33	10	0
Held-out: unused-input traps	16	9	7	0
Verified natural-language questions (physics-questions-1)	1567	478	1086	0

Wrong means a number that disagrees with the verified answer. On the verified-question set, 3 of 481 answers differ from the record: two are unit spellings (ohm/ohms) or a self-contradictory question ('angular frequency in Hz'), one pairs two records that share an id; none is a reading error. Refusals are expected for questions outside the certified capabilities or with unusual wording; solve() covers every capability with typed inputs.

Optional decision model (auto (gated) + suggest)

This repository also holds a compact ONNX decision model (onnx/, 37.4 MB int8, 33.4M parameters, CPU; fine-tuned from BAAI/bge-small-en-v1.5): it reads a question the rule-based reader refuses, names the capability it needs and copies the input values from the words (it never writes a digit itself).

When it answers by itself: only through a gate — confidence at or above the zero-wrong cut chosen on validation with this very file, every required input read, every number in the question used, and the engine's verification; the asked quantity read by the rule-based reader (never inferred from which value is missing), every input copied with its unit, no direction or sign word in the question, and the relation's context conditions. The answer says it was read by the model. Otherwise it suggests the capability and a pre-filled call for you to check.

Measured (held-out questions the rule-based reader refuses)	Value
Right capability named	157 of 159 named right (227 held-out questions the rule-based reader refuses)
Input values copied	through the gate: 19 answered automatically on test, 0 wrong; 201 get a suggestion
CPU latency, p95	47.2 ms (4 threads)

Through the gate it answered 19 held-out test and 22 validation questions that the rule-based reader refuses, with 0 wrong. On 1,377 rejected teacher questions it answered 41 distinct questions by itself, all correct by hand calculation (in every disagreement the teacher's key was wrong), and the 88 benchmark questions the reader refuses stay refused.

pip install "physics-ai999x[onnx]"
hf download AI999X/physics-ai999x --include "onnx/*" --local-dir physics_model

import physics_ai999x as physics
reader = physics.DecisionReader("physics_model/onnx")
physics.ask("A 0.25 m rod generates 5 V while sliding in a 1 T field. How fast is the rod moving?", reader=reader).value   # -> 20.0 (m/s), read by the model, verified by the engine
physics.ask("quick one: spring 500 N/m, mass 0.01 kg, amplitude 0.004 m, top speed?", reader=reader).message
# -> ... Suggestion (decision model, not verified): this looks like `shm_max_speed` ... run physics_ai999x.shm_max_speed(k='500 N/m', m='0.01 kg', X='0.004 m')

Why the gate: the engine verifies a computation, not a reading. A copied number carries no sign, and the model does not decide what is asked — the rule-based reader names the asked quantity, and a question that sets directions against each other goes to a suggestion. Before that check, one held-out question ('downward at 2.0 m/s ... upward acceleration') was answered 8 m/s instead of 4 m/s upward.

ONNX formula pack (deterministic, no Python needed)

onnx/formulas/ holds 365 tiny ONNX graphs (94.7 KB in total): every certified closed-form capability compiled from its exact formula — not learned, so there is nothing approximate in them. Float64, standard operators only; run them with onnxruntime in Python, C++, C#, Java or in the browser (onnxruntime-web).

Verified against the Python engine: 13,256 of 13,256 checks agree, worst relative error 9.9e-15. onnx/formulas/formulas.json lists each graph's inputs (SI units), outputs and, where a formula has several solution branches, which one to take.

import numpy as np, onnxruntime as ort
s = ort.InferenceSession("onnx/formulas/uniform_motion__d.onnx")
print(s.run(None, {"v": np.array(2.0), "t": np.array(2.0)}))

import * as ort from "onnxruntime-web";
const s = await ort.InferenceSession.create("onnx/formulas/uniform_motion__d.onnx");
const out = await s.run({ v: new ort.Tensor("float64", new Float64Array([2.0]), []), t: new ort.Tensor("float64", new Float64Array([2.0]), []) });

Find the exact formula, never guess (autocomplete): onnx/formulas/catalog.json is a search index of every graph (names, keywords, the unknown it solves for, inputs with SI units, LaTeX). formula-pack.js searches it, describes an entry and runs the exact graph with the branch rule; formula-pack.d.ts gives your editor every formula id and input name.

import { loadPack } from "./formula-pack.js";
const pack = await loadPack("https://huggingface.co/AI999X/physics-ai999x/resolve/main/onnx/formulas");
pack.search("kinetic energy speed");   // best match first: the formula and the unknown it solves for

Guidebook: docs/physics_ai999x_onnx_formulas.md (also HTML and PDF) documents every graph — LaTeX, inputs and units, branches, conditions — each with an example executed on the ONNX file and compared with the Python engine.

Reading questions in words stays in the Python package (and the optional decision model); the formula pack is the compute half, for apps, calculators and edge devices.

Capabilities

23 certified derivations, 16 certified model_familys, 2 certified procedures, 107 certified relations. A few of them:

• Derivations: acceleration with friction level; angular acceleration from tangential; angular frequency period; average speed equal instantaneous; centripetal acceleration from period; dielectric increases capacitance; displacement from velocity time; distance fallen time; energy via resistance; final speed from work; focal length; max height from velocity displacement; net work to speed; period from wave speed; power from voltage and resistance; solve for g; speed after height; speed after time dropped; speed from uniform motion; temp change inverse mass; time to stop; work to speed rest; zero initial speed
• Model_familys: A gas between two states; Apparent weight in an elevator; Block on a surface pulled over a pulley; Collisions along a line; Doppler effect for sound; Ideal banking of a curve; Maximum speed on a flat curve; Mixing to thermal equilibrium; Motion in stages; Projectile motion; RC circuits; Radioactive decay; Recoil from rest; Rolling from a height; Series and parallel resistor and capacitor networks; Two masses over a pulley
• Procedures: Two resistors in series and in parallel; Velocity and displacement after constant acceleration
• Relations: Absolute pressure at a depth in a fluid; Acceleration down a frictionless incline; Angular frequency and frequency; Angular momentum of a point mass on a circle; Angular momentum of a rotating body; Angular velocity and angle turned under constant angular acceleration; Angular velocity under constant angular acceleration; Average net force from a change of momentum; Average power; Capacitance of parallel plates; Carnot (maximum) efficiency; Centripetal acceleration; Centripetal acceleration from angular velocity; Centripetal force; Charge carried by a steady current; Charge on a capacitor; Coefficient of performance of a heat pump; Coefficient of performance of a refrigerator; Component of the weight along an incline; Current dying away in an RL circuit; De Broglie wavelength; Displacement from the mean of initial and final velocity; Displacement under constant acceleration; Efficiency of a heat engine …

physics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 109 formulas

Show every formula, by topic

AC circuits

Reactance of a capacitor

$$X=\frac{1}{2\pi fC}X\; =\; \backslash frac\{1\}\{2\; \backslash pi\; f\; C\}$$

Reactance of an inductor

$$X=2\pi fLX\; =\; 2\; \backslash pi\; f\; L$$

Resonant frequency of an LC circuit

$$f=\frac{1}{2\pi \sqrt{LC}}f\; =\; \backslash frac\{1\}\{2\; \backslash pi\; \backslash sqrt\{L\; C\}\}$$

Rms current of a sinusoidal supply

$$I_{\text{rms}}=\frac{I_0}{\sqrt{2}}I\_\{\backslash text\{rms\}\}\; =\; \backslash frac\{I\_\{0\}\}\{\backslash sqrt\{2\}\}$$

Rms voltage of a sinusoidal supply

$$V_{\text{rms}}=\frac{V_0}{\sqrt{2}}V\_\{\backslash text\{rms\}\}\; =\; \backslash frac\{V\_\{0\}\}\{\backslash sqrt\{2\}\}$$

Capacitors

Capacitance of parallel plates

$$C=\frac{{\epsilon }_{0}A}{d}C\; =\; \backslash frac\{\backslash varepsilon\_\{0\}\; A\}\{d\}$$

Charge on a capacitor

$$Q=CVQ\; =\; C\; V$$

Energy stored in a capacitor

$$\text{Ecap}=\frac{C{V}^2}{2}\backslash text\{Ecap\}\; =\; \backslash frac\{C\; V^\{2\}\}\{2\}$$

Circuits

Power from current and resistance

$$P=I^{2}RP\; =\; I^\{2\}\; R$$

Power from voltage and resistance

$$P=\frac{V^2}{R}P\; =\; \backslash frac\{V^\{2\}\}\{R\}$$

Resistance of a uniform wire

$$R=\frac{{\rho }_{e}L}{A}R\; =\; \backslash frac\{\backslash rho\_\{e\}\; L\}\{A\}$$

Two resistors in series and in parallel

$$r_s=r_1+r_2,r_p(r_1+r_2)=r_1{r}_{2}r\_\{s\}\; =\; r\_\{1\}\; +\; r\_\{2\},\backslash quad\; r\_\{p\}\; \backslash left(r\_\{1\}\; +\; r\_\{2\}\backslash right)\; =\; r\_\{1\}\; r\_\{2\}$$

Circular

Angular frequency and frequency

$$\omega =2\pi f\backslash omega\; =\; 2\; \backslash pi\; f$$

Centripetal acceleration

$$a=\frac{v^2}{r}a\; =\; \backslash frac\{v^\{2\}\}\{r\}$$

Centripetal force

$$F=\frac{m{v}^2}{r}F\; =\; \backslash frac\{m\; v^\{2\}\}\{r\}$$

Period and frequency

$$T=\frac{1}{f}T\; =\; \backslash frac\{1\}\{f\}$$

Speed on a circle from angular speed

$$v=\omega rv\; =\; \backslash omega\; r$$

Dynamics

Acceleration down a frictionless incline

$$a=g\sin \left(\theta \right)a\; =\; g\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Average net force from a change of momentum

$$F=\frac{p_2-p_1}{t}F\; =\; \backslash frac\{p\_2\; -\; p\_1\}\{t\}$$

Component of the weight along an incline

$$F=mg\sin \left(\theta \right)F\; =\; m\; g\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Impulse of a constant force

$$J=FtJ\; =\; F\; t$$

Linear momentum

$$p=mvp\; =\; m\; v$$

Net force and acceleration

$$F=maF\; =\; m\; a$$

Net force with an opposing friction force

$$F=P-fF\; =\; P\; -\; f$$

Normal force on a level surface

$$N=WN\; =\; W$$

Normal force on an incline

$$N=mg\cos \left(\theta \right)N\; =\; m\; g\; \backslash cos\backslash left(\backslash theta\backslash right)$$

Sliding friction

$$f=\mu Nf\; =\; \backslash mu\; N$$

Tension in a rope holding a hanging body

$$T=mgT\; =\; m\; g$$

Weight in a uniform gravitational field

$$W=mgW\; =\; m\; g$$

Electricity

Charge carried by a steady current

$$Q=ItQ\; =\; I\; t$$

Electric power

$$P=VIP\; =\; V\; I$$

Voltage across an ohmic resistance

$$V=IRV\; =\; I\; R$$

Electrostatics

Electric field of a point charge

$$E_f=\frac{k_{e}Q}{r^2}E\_\{f\}\; =\; \backslash frac\{k\_\{e\}\; Q\}\{r^\{2\}\}$$

Electric potential of a point charge

$$V=\frac{k_{e}Q}{r}V\; =\; \backslash frac\{k\_\{e\}\; Q\}\{r\}$$

Force on a charge in an electric field

$$F=q{E}_{f}F\; =\; q\; E\_\{f\}$$

Potential difference in a uniform field

$$V=E_{f}dV\; =\; E\_\{f\}\; d$$

Energy

Average power

$$P=\frac{E}{t}P\; =\; \backslash frac\{E\}\{t\}$$

Energy stored in a linear spring

$$U=\frac{k{x}^2}{2}U\; =\; \backslash frac\{k\; x^\{2\}\}\{2\}$$

Force of a linear spring

$$F=kxF\; =\; k\; x$$

Gravitational potential energy near the surface

$$U=mghU\; =\; m\; g\; h$$

Power of a force on a moving body

$$P=FvP\; =\; F\; v$$

Speed of a body launched by a spring

$$\frac{k{x}^2}{2}=\frac{m{v}^2}{2}\backslash frac\{k\; x^\{2\}\}\{2\}\; =\; \backslash frac\{m\; v^\{2\}\}\{2\}$$

Translational kinetic energy

$$K=\frac{m{v}^2}{2}K\; =\; \backslash frac\{m\; v^\{2\}\}\{2\}$$

Work by a constant force

$$W=Fd\cos \left(\theta \right)W\; =\; F\; d\; \backslash cos\backslash left(\backslash theta\backslash right)$$

Work-energy theorem

$$W=\frac{m(v^2-v_0^2)}{2}W\; =\; \backslash frac\{m\; \backslash left(v^\{2\}\; -\; v\_0^\{2\}\backslash right)\}\{2\}$$

Fluids

Absolute pressure at a depth in a fluid

$$P=P_0+\rho ghP\; =\; P\_\{0\}\; +\; \backslash rho\; g\; h$$

Flow rate as volume per time

$$Q=\frac{V}{t}Q\; =\; \backslash frac\{V\}\{t\}$$

Flow rate through a cross section

$$Q=AvQ\; =\; A\; v$$

Mass density

$$\rho =\frac{m}{V}\backslash rho\; =\; \backslash frac\{m\}\{V\}$$

Pressure of a column of fluid

$$P=\rho ghP\; =\; \backslash rho\; g\; h$$

Pressure of a force on an area

$$P=\frac{F}{A}P\; =\; \backslash frac\{F\}\{A\}$$

Gravitation

Gravitational acceleration at a distance from a mass

$$g=\frac{GM}{r^2}g\; =\; \backslash frac\{G\; M\}\{r^\{2\}\}$$

Period of a circular orbit

$$T=2\pi \sqrt{\frac{r^3}{GM}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{r^\{3\}\}\{G\; M\}\}$$

Speed in a circular orbit

$$v=\sqrt{\frac{GM}{r}}v\; =\; \backslash sqrt\{\backslash frac\{G\; M\}\{r\}\}$$

Induction

Current dying away in an RL circuit

$$I=I_0{e}^{\frac{-t}{\tau }}I\; =\; I\_\{0\}\; e^\{\backslash frac\{-t\}\{\backslash tau\}\}$$

Emf of a rod moving through a magnetic field

$$\text{emf}=Blv\backslash text\{emf\}\; =\; B\; l\; v$$

Time constant of an RL circuit

$$\tau =\frac{L}{R}\backslash tau\; =\; \backslash frac\{L\}\{R\}$$

Kinematics

Displacement from the mean of initial and final velocity

$$x=\frac{(v_0+v)t}{2}x\; =\; \backslash frac\{\backslash left(v\_0\; +\; v\backslash right)\; t\}\{2\}$$

Displacement under constant acceleration

$$x=v_{0}t+\frac{a{t}^2}{2}x\; =\; v\_0\; t\; +\; \backslash frac\{a\; t^\{2\}\}\{2\}$$

Maximum height of a vertical throw

$$h=\frac{v_0^2}{2g}h\; =\; \backslash frac\{v\_0^\{2\}\}\{2\; g\}$$

Motion at constant velocity

$$d=vtd\; =\; v\; t$$

Speed after a displacement under constant acceleration

$$v^2=v_0^2+2axv^\{2\}\; =\; v\_0^\{2\}\; +\; 2\; a\; x$$

Speed after falling from rest

$$v=\sqrt{2gh}v\; =\; \backslash sqrt\{2\; g\; h\}$$

Time to the top of a vertical throw

$$t=\frac{v_0}{g}t\; =\; \backslash frac\{v\_0\}\{g\}$$

Velocity and displacement after constant acceleration

$$v=u+at,2s=(u+v)tv\; =\; u\; +\; a\; t,\backslash quad\; 2\; s\; =\; \backslash left(u\; +\; v\backslash right)\; t$$

Velocity under constant acceleration

$$v=v_0+atv\; =\; v\_0\; +\; a\; t$$

Magnetism

Magnetic field at the centre of a circular loop

$$B=\frac{{\mu }_{0}I}{2r}B\; =\; \backslash frac\{\backslash mu\_\{0\}\; I\}\{2\; r\}$$

Magnetic field inside a long solenoid

$$B={\mu }_0{n}_{t}IB\; =\; \backslash mu\_\{0\}\; n\_\{t\}\; I$$

Magnetic field of a long straight wire

$$B=\frac{{\mu }_{0}I}{2\pi r}B\; =\; \backslash frac\{\backslash mu\_\{0\}\; I\}\{2\; \backslash pi\; r\}$$

Magnetic force on a current-carrying wire

$$F=ILB\sin \left(\theta \right)F\; =\; I\; L\; B\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Magnetic force on a moving charge

$$F=qvB\sin \left(\theta \right)F\; =\; q\; v\; B\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Radius of a charged particle's circle in a magnetic field

$$r=\frac{mv}{qB\sin \left(\theta \right)}r\; =\; \backslash frac\{m\; v\}\{q\; B\; \backslash sin\backslash left(\backslash theta\backslash right)\}$$

Optics

Focal length of a spherical mirror

$$f=\frac{R}{2}f\; =\; \backslash frac\{R\}\{2\}$$

Index of refraction

$$n=\frac{c}{v}n\; =\; \backslash frac\{c\}\{v\}$$

Thin lens equation

$$\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}\backslash frac\{1\}\{d\_\{o\}\}\; +\; \backslash frac\{1\}\{d\_\{i\}\}\; =\; \backslash frac\{1\}\{f\}$$

Wavelength of light in a medium

$${\lambda }_n=\frac{\lambda }{n}\backslash lambda\_\{n\}\; =\; \backslash frac\{\backslash lambda\}\{n\}$$

Oscillations

Maximum speed of a mass on a spring

$$v=X\sqrt{\frac{k}{m}}v\; =\; X\; \backslash sqrt\{\backslash frac\{k\}\{m\}\}$$

Period of a mass on a spring

$$T=2\pi \sqrt{\frac{m}{k}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{m\}\{k\}\}$$

Period of a simple pendulum

$$T=2\pi \sqrt{\frac{L_p}{g}}T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{L\_\{p\}\}\{g\}\}$$

Quantum

De Broglie wavelength

$$\lambda =\frac{h}{p}\backslash lambda\; =\; \backslash frac\{h\}\{p\}$$

Energy of a photon

$$E=hfE\; =\; h\; f$$

Momentum of a photon

$$p=\frac{h}{\lambda }p\; =\; \backslash frac\{h\}\{\backslash lambda\}$$

Photon energy from wavelength

$$E=\frac{hc}{\lambda }E\; =\; \backslash frac\{h\; c\}\{\backslash lambda\}$$

RC circuits

Time constant of an RC circuit

$$\tau =RC\backslash tau\; =\; R\; C$$

Voltage of a charging capacitor

$$V=\text{emf}(1-e^{\frac{-t}{RC}})V\; =\; \backslash text\{emf\}\; \backslash left(1\; -\; e^\{\backslash frac\{-t\}\{R\; C\}\}\backslash right)$$

Voltage of a discharging capacitor

$$V=V_0{e}^{\frac{-t}{RC}}V\; =\; V\_\{0\}\; e^\{\backslash frac\{-t\}\{R\; C\}\}$$

Relativity

Rest energy

$$E_0=m{c}^{2}E\_\{0\}\; =\; m\; c^\{2\}$$

Rotation

Angular momentum of a point mass on a circle

$$L=mvrL\; =\; m\; v\; r$$

Angular momentum of a rotating body

$$L=I_r\omega L\; =\; I\_\{r\}\; \backslash omega$$

Angular velocity and angle turned under constant angular acceleration

$${\omega }^2={\omega }_0^2+2\alpha \theta \backslash omega^\{2\}\; =\; \backslash omega\_\{0\}^\{2\}\; +\; 2\; \backslash alpha\; \backslash theta$$

Angular velocity under constant angular acceleration

$$\omega ={\omega }_0+\alpha t\backslash omega\; =\; \backslash omega\_\{0\}\; +\; \backslash alpha\; t$$

Centripetal acceleration from angular velocity

$$a=r{\omega }^{2}a\; =\; r\; \backslash omega^\{2\}$$

Rotational kinetic energy

$$K_R=\frac{I_r{\omega }^2}{2}K\_\{R\}\; =\; \backslash frac\{I\_\{r\}\; \backslash omega^\{2\}\}\{2\}$$

Tangential acceleration of a point on a rotating body

$$a=r\alpha a\; =\; r\; \backslash alpha$$

Torque of a force

$$\tau =rF\sin \left(\theta \right)\backslash tau\; =\; r\; F\; \backslash sin\backslash left(\backslash theta\backslash right)$$

Thermal

Carnot (maximum) efficiency

$$\text{eff}=1-\frac{T_c}{T_h}\backslash text\{eff\}\; =\; 1\; -\; \backslash frac\{T\_\{c\}\}\{T\_\{h\}\}$$

Coefficient of performance of a heat pump

$$\text{cop}=\frac{Q_h}{W}\backslash text\{cop\}\; =\; \backslash frac\{Q\_\{h\}\}\{W\}$$

Coefficient of performance of a refrigerator

$$\text{cop}=\frac{Q_c}{W}\backslash text\{cop\}\; =\; \backslash frac\{Q\_\{c\}\}\{W\}$$

Efficiency of a heat engine

$$\text{eff}=\frac{W}{Q_h}\backslash text\{eff\}\; =\; \backslash frac\{W\}\{Q\_\{h\}\}$$

Energy balance of a cyclic heat engine

$$W=Q_h-Q_{c}W\; =\; Q\_\{h\}\; -\; Q\_\{c\}$$

Heat for a temperature change

$$Q=mc\mathrm{\Delta }TQ\; =\; m\; c\; \backslash Delta\; T$$

Heat to boil or condense

$$Q=m{L}_{v}Q\; =\; m\; L\_\{v\}$$

Heat to melt or freeze

$$Q=m{L}_{f}Q\; =\; m\; L\_\{f\}$$

Ideal gas law

$$PV=nRTP\; V\; =\; n\; R\; T$$

Linear thermal expansion

$$d_L={\alpha }_{t}L{d}_{T}d\_\{L\}\; =\; \backslash alpha\_\{t\}\; L\; d\_\{T\}$$

Thermal expansion of an area

$$d_A=2{\alpha }_{t}A{d}_{T}d\_\{A\}\; =\; 2\; \backslash alpha\_\{t\}\; A\; d\_\{T\}$$

Thermal expansion of volume

$$d_V=\beta V_0{d}_{T}d\_\{V\}\; =\; \backslash beta\; V\_\{0\}\; d\_\{T\}$$

Waves

Intensity as power per area

$$I=\frac{P}{A}I\; =\; \backslash frac\{P\}\{A\}$$

Speed of a periodic wave

$$v=f\lambda v\; =\; f\; \backslash lambda$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified physics capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (kinematics, forces and Newton's laws, work and energy, momentum and collisions, circular motion and gravitation, rotation, oscillations and waves, fluids, heat and thermodynamics, electricity and circuits, magnetism and induction, optics and modern physics); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{physics_ai999x_2_0_0,
  title  = {physics-ai999x: an offline, verified physics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {2.0.0},
  url    = {https://huggingface.co/AI999X/physics-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/physics-ai999x/discussions or visit https://ai999x.com.

Support the project

physics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com